We note that given \(\mathcal {H} \subseteq 2^Z\) , determining the expectation-threshold or fractional expectation-threshold is a difficult task in general. In particular, the fractional expectation-threshold is the solution to a linear program with variables corresponding to the collection of subsets in \(\mathcal {H}\) , which for many applications is of exponential size. (The expectation-threshold is in general an integer linear program.) Nevertheless, the results of [1]}, [2]} immediately imply several previously difficult results including the threshold for containing a perfect matching in a random hypergraph (due to Johansson, Kahn, and Vu [3]}) or containing a given bounded-degree spanning tree in a random graph (due to Montgomery [4]}). A crucial factor in these applications is that the uniform distribution has optimal spread. This follows, for example, from the fact that vertex permutations act transitively on the objects in question (see [1]}). In contrast, in the minimum-degree setting using vertex permutations is a non-starter, and it is not obvious that the uniform distribution has sufficiently small spread to find the thresholds. The focus of this work is on providing methods for constructing spread measures in situations where one cannot rely on the “spread from vertex permutations”. A similar difficulty is present in work on the threshold for containing a Steiner triple system [6]}, [7]}, where the connection between spread and thresholds was exploited alongside other tools for understanding Steiner systems and Latin squares.
